Problem 5 - Beginning and Intermediate Algebra

Question

Solve. $$\frac{1}{2}+\frac{2}{x}=\frac{1}{3}+\frac{3}{x}$$

Step-by-Step Solution

Answer

The short version of the answer is: To solve the equation \(\frac{1}{2}+\frac{2}{x}=\frac{1}{3}+\frac{3}{x}\), find the common denominator (6x), multiply both sides by the LCD, simplify the equation, and solve for x. After these steps, we find that the solution is \(x=6\).

Step 1: Find the common denominator of the fractions

In this case, the fractions are \(\frac{1}{2}, \ \frac{2}{x}, \ \frac{1}{3}, \ \text{and} \ \frac{3}{x}\). The least common denominator (LCD) should be a multiple of all the denominators. Here, the LCD will be 6x, since it is a multiple of both 2, 3, and x.

Step 2: Multiply both sides of the equation by the LCD

To clear the fractions, multiply both sides of the equation by 6x: \[(6x)\left(\frac{1}{2}+\frac{2}{x}\right)=(6x)\left(\frac{1}{3}+\frac{3}{x}\right)\]

Step 3: Simplify both sides of the equation

Distribute the 6x on both sides of the equation and simplify: \[3x+12=2x+18\]

Step 4: Solve for x

Subtract 2x from both sides to isolate x: \[x+12=18\]. Now, subtract 12 from both sides: \[x=6\]

Step 5: Check the solution

To ensure the solution is correct, substitute x=6 back into the original equation: \[\frac{1}{2}+\frac{2}{6}=\frac{1}{3}+\frac{3}{6}\]. Simplifying gives: \[\frac{1}{2}+\frac{1}{3}=\frac{1}{3}+\frac{1}{2}\] Since the left side of the equation equals the right side, x=6 is the correct solution.

Key Concepts

Least Common Denominator

When solving equations involving fractions, finding the least common denominator (LCD) is a crucial step. The LCD is the smallest number that is a multiple of all the denominators in the equation. This enables you to eliminate the fractions and simplify the equation.

Finding the LCD involves:

• Identifying the denominators in the problem. In our exercise, the denominators were 2, 3, and x.
• Finding the smallest number that can be divided evenly by each of these denominators. For 2 and 3, this is 6. Including x, the LCD becomes 6x.

By using the LCD, you can convert the equation into one without fractions, making it easier to solve.

Distributive Property

The distributive property is a valuable tool in algebra that involves distributing a factor over a sum or difference inside parentheses. It's expressed as \(a(b + c) = ab + ac\). This property helps expand and simplify expressions.

In our exercise:

• Once you have the LCD, you multiply it by each term in the equation using the distributive property.
• This helps clear out the fractions, transforming the equation into one that is easier to solve.
• For example, multiplying \(6x\) by the left side of the equation \((\frac{1}{2} + \frac{2}{x})\), results in \(3x + 12\).
• Apply the same process to the right side of the equation.

This step is essential for reducing complexity and solving for the variable.

Linear Equations

Linear equations are equations of the first degree, meaning they have no powers higher than one. They take the simplest algebraic form that can be solved using basic operations.

The equation derived in the exercise, \(3x + 12 = 2x + 18\), is a linear equation. The characteristics include:

• It involves variables raised only to the first power.
• Simplifying and solving it involves balancing and isolating terms on either side.
• Operations include addition, subtraction, multiplication, and division.

Solving linear equations typically starts by simplifying and then isolating the variable to determine its value.

Substitution Method

The substitution method is a technique used to confirm the solution of an equation. It involves plugging the found solution back into the original equation to ensure that both sides equal.

For the exercise, once we determined that \(x = 6\), we substituted back to check:

• Replace \(x\) with 6 in the original equation: \(\frac{1}{2} + \frac{2}{6} = \frac{1}{3} + \frac{3}{6}\).
• Simplify both sides to check if they are equal: \(\frac{1}{2} + \frac{1}{3}\) should equal \(\frac{1}{3} + \frac{1}{2}\).
• Both sides of the equation should simplify to the same value, confirming \(x = 6\) is correct.

The substitution method ensures that the solution satisfies the initial equation, providing a reliable and accurate verification.